The concepts of random error and reliability of measurements that are familiar in traditional theories based on the notions of “true values” and “errors” can be represented by a probability model having a simpler formal structure and fewer special assumptions about random sampling and independence of measurements. In this model formulas that relate observable events are derived from probability axioms and from primitive terms that refer to observable events, without an intermediate structure containing variances and correlations of “true” and “error” components of scores. While more economical in language and formalism, the model at the same time is more general than classical theories and applies to stochastic processes in which joint distributions of many dependent random variables are of interest. In addition, it clarifies some long-standing problems concerning “experimental independence” of measurements and the relation of sampling of individuals to sampling of measurements.